Subadditive ergodic theorems for random sets in infinite dimensions
Subadditive ergodic theorems for superstationary families of compact convex random subsets of a real Banach space are derived. The classical strong law of large numbers for independent identically distributed random variables (due to Kolmogorov) has been extended into several directions. One possibility is to consider stationary and even superstationary families of random variables; see, e.g., \textit{U. Krengel} [C. R. Acad. Sci., Paris, Sér. A 282, 1019-1021 (1976; Zbl 0343.60017)]. Another possibility is to relax the underlying additivity condition. This has been done e.g. by \textit{J. F. C. Kingman} [J. R. Stat. Soc., Ser. B 30, 499-510 (1968; Zbl 0182.22802)] and the reviewer [Ann. Probab. 19, No. 4, 1575-1586 (1991; Zbl 0748.60033)] in which subadditive and, more generally, almost subadditive families of random variables are considered. Based on these (and some other) results and using e.g. a distance function instead of support functions, the authors obtain very general ergodic theorems for random subsets of Banach spaces extending earlier results due to \textit{Z. Artstein} and \textit{R. Vitale} [ibid. 3, 879-882 (1975; Zbl 0313.60012)], \textit{M. Abid} [C. R. Acad. Sci., Paris, Sér. A 287, 149-152 (1978; Zbl 0386.60028)] and the reviewer [Z. Wahrscheinlichkeitstheorie Verw. Geb. 62, 125-135 (1983; Zbl 0489.60005)].
scientific article; zbMATH DE number 1556863